Regular representations and group algebras

The complex group algebra has one basis symbol for each group element. Multiply basis symbols using the group law, then distribute over sums and scalar coefficients. The identity basis symbol is the multiplicative identity. Left multiplication by a group element permutes this basis and defines the left regular representation. A nonidentity element cannot fix any basis label, since cancelling that label would force the translating element to be the identity. The identity operator fixes the entire basis.

Consequently the regular character is the group order at the identity and zero elsewhere. Its inner product with an irreducible character is the dimension of that irreducible. This proves every irreducible occurs, rather than assuming that a previously found list is complete. Comparing dimensions gives the sum of squared irreducible dimensions equal to the group order. Counting dimensions only once would miss the multiplicity built into the regular representation; a two-dimensional irreducible contributes four dimensions here.

Every representation extends linearly to an action of the group algebra: a linear combination of group elements acts by the same linear combination of their matrices. Applying all inequivalent irreducible representations at once gives an algebra homomorphism into a direct sum of full endomorphism algebras. Over the complex numbers it is an isomorphism. One route uses semisimplicity and the structure theorem for finite-dimensional semisimple complex algebras; the regular multiplicities then identify the sizes of its matrix blocks.

The field restriction is significant. Maschke's complete reducibility works over any field with invertible group order, but the simple algebra blocks over a general field can involve nontrivial division algebras rather than just matrices over that field. In characteristic dividing the order, even semisimplicity can fail. We therefore state the full matrix-block decomposition over the complex numbers here. It says the group algebra has exactly enough independent matrix entries to encode an arbitrary function on the finite group, anticipating Fourier inversion.

A central idempotent is an algebra element whose square equals itself and which commutes with every algebra element. A primitive central idempotent associated with one irreducible type projects onto its invariant isotypic component: the sum of all copies of that type. An arbitrary central idempotent can select several whole isotypic components at once. Even a primitive one does not isolate a single copy when that type has multiplicity greater than one. For the trivial type, averaging all group elements gives the projection onto fixed vectors. Multiplying by another average leaves it unchanged because every group element appears equally often.

For a general irreducible type, the coefficient of each group element in its central idempotent is the dimension times the character of the inverse element divided by the group order. Centrality follows from the character being constant on conjugacy classes. Schur's lemma makes its action scalar on each irreducible; taking traces and applying character orthogonality makes that scalar one for the selected type and zero for all others. Distinct such idempotents multiply to zero, and their sum is the identity.